Properties

Label 2025.2.b.g.649.3
Level $2025$
Weight $2$
Character 2025.649
Analytic conductor $16.170$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2025 = 3^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2025.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(16.1697064093\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 405)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 649.3
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 2025.649
Dual form 2025.2.b.g.649.2

$q$-expansion

\(f(q)\) \(=\) \(q+0.732051i q^{2} +1.46410 q^{4} -4.73205i q^{7} +2.53590i q^{8} +O(q^{10})\) \(q+0.732051i q^{2} +1.46410 q^{4} -4.73205i q^{7} +2.53590i q^{8} -5.73205 q^{11} -1.46410i q^{13} +3.46410 q^{14} +1.07180 q^{16} -2.73205i q^{17} -4.46410 q^{19} -4.19615i q^{22} +3.46410i q^{23} +1.07180 q^{26} -6.92820i q^{28} -3.19615 q^{29} -3.00000 q^{31} +5.85641i q^{32} +2.00000 q^{34} -2.73205i q^{37} -3.26795i q^{38} -7.19615 q^{41} -0.196152i q^{43} -8.39230 q^{44} -2.53590 q^{46} -8.73205i q^{47} -15.3923 q^{49} -2.14359i q^{52} -6.73205i q^{53} +12.0000 q^{56} -2.33975i q^{58} +8.26795 q^{59} +4.00000 q^{61} -2.19615i q^{62} -2.14359 q^{64} +3.46410i q^{67} -4.00000i q^{68} -3.73205 q^{71} +7.66025i q^{73} +2.00000 q^{74} -6.53590 q^{76} +27.1244i q^{77} -15.4641 q^{79} -5.26795i q^{82} -2.19615i q^{83} +0.143594 q^{86} -14.5359i q^{88} -5.19615 q^{89} -6.92820 q^{91} +5.07180i q^{92} +6.39230 q^{94} -9.66025i q^{97} -11.2679i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{4} - 16 q^{11} + 32 q^{16} - 4 q^{19} + 32 q^{26} + 8 q^{29} - 12 q^{31} + 8 q^{34} - 8 q^{41} + 8 q^{44} - 24 q^{46} - 20 q^{49} + 48 q^{56} + 40 q^{59} + 16 q^{61} - 64 q^{64} - 8 q^{71} + 8 q^{74} - 40 q^{76} - 48 q^{79} + 56 q^{86} - 16 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2025\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(1702\)
\(\chi(n)\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.732051i 0.517638i 0.965926 + 0.258819i \(0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(3\) 0 0
\(4\) 1.46410 0.732051
\(5\) 0 0
\(6\) 0 0
\(7\) − 4.73205i − 1.78855i −0.447521 0.894274i \(-0.647693\pi\)
0.447521 0.894274i \(-0.352307\pi\)
\(8\) 2.53590i 0.896575i
\(9\) 0 0
\(10\) 0 0
\(11\) −5.73205 −1.72828 −0.864139 0.503253i \(-0.832136\pi\)
−0.864139 + 0.503253i \(0.832136\pi\)
\(12\) 0 0
\(13\) − 1.46410i − 0.406069i −0.979172 0.203034i \(-0.934920\pi\)
0.979172 0.203034i \(-0.0650803\pi\)
\(14\) 3.46410 0.925820
\(15\) 0 0
\(16\) 1.07180 0.267949
\(17\) − 2.73205i − 0.662620i −0.943522 0.331310i \(-0.892509\pi\)
0.943522 0.331310i \(-0.107491\pi\)
\(18\) 0 0
\(19\) −4.46410 −1.02414 −0.512068 0.858945i \(-0.671120\pi\)
−0.512068 + 0.858945i \(0.671120\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 4.19615i − 0.894623i
\(23\) 3.46410i 0.722315i 0.932505 + 0.361158i \(0.117618\pi\)
−0.932505 + 0.361158i \(0.882382\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 1.07180 0.210197
\(27\) 0 0
\(28\) − 6.92820i − 1.30931i
\(29\) −3.19615 −0.593511 −0.296755 0.954954i \(-0.595905\pi\)
−0.296755 + 0.954954i \(0.595905\pi\)
\(30\) 0 0
\(31\) −3.00000 −0.538816 −0.269408 0.963026i \(-0.586828\pi\)
−0.269408 + 0.963026i \(0.586828\pi\)
\(32\) 5.85641i 1.03528i
\(33\) 0 0
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) 0 0
\(37\) − 2.73205i − 0.449146i −0.974457 0.224573i \(-0.927901\pi\)
0.974457 0.224573i \(-0.0720988\pi\)
\(38\) − 3.26795i − 0.530131i
\(39\) 0 0
\(40\) 0 0
\(41\) −7.19615 −1.12385 −0.561925 0.827188i \(-0.689939\pi\)
−0.561925 + 0.827188i \(0.689939\pi\)
\(42\) 0 0
\(43\) − 0.196152i − 0.0299130i −0.999888 0.0149565i \(-0.995239\pi\)
0.999888 0.0149565i \(-0.00476097\pi\)
\(44\) −8.39230 −1.26519
\(45\) 0 0
\(46\) −2.53590 −0.373898
\(47\) − 8.73205i − 1.27370i −0.770988 0.636850i \(-0.780237\pi\)
0.770988 0.636850i \(-0.219763\pi\)
\(48\) 0 0
\(49\) −15.3923 −2.19890
\(50\) 0 0
\(51\) 0 0
\(52\) − 2.14359i − 0.297263i
\(53\) − 6.73205i − 0.924718i −0.886693 0.462359i \(-0.847003\pi\)
0.886693 0.462359i \(-0.152997\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 12.0000 1.60357
\(57\) 0 0
\(58\) − 2.33975i − 0.307224i
\(59\) 8.26795 1.07640 0.538198 0.842819i \(-0.319105\pi\)
0.538198 + 0.842819i \(0.319105\pi\)
\(60\) 0 0
\(61\) 4.00000 0.512148 0.256074 0.966657i \(-0.417571\pi\)
0.256074 + 0.966657i \(0.417571\pi\)
\(62\) − 2.19615i − 0.278912i
\(63\) 0 0
\(64\) −2.14359 −0.267949
\(65\) 0 0
\(66\) 0 0
\(67\) 3.46410i 0.423207i 0.977356 + 0.211604i \(0.0678686\pi\)
−0.977356 + 0.211604i \(0.932131\pi\)
\(68\) − 4.00000i − 0.485071i
\(69\) 0 0
\(70\) 0 0
\(71\) −3.73205 −0.442913 −0.221456 0.975170i \(-0.571081\pi\)
−0.221456 + 0.975170i \(0.571081\pi\)
\(72\) 0 0
\(73\) 7.66025i 0.896565i 0.893892 + 0.448282i \(0.147964\pi\)
−0.893892 + 0.448282i \(0.852036\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) −6.53590 −0.749719
\(77\) 27.1244i 3.09111i
\(78\) 0 0
\(79\) −15.4641 −1.73985 −0.869924 0.493186i \(-0.835832\pi\)
−0.869924 + 0.493186i \(0.835832\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) − 5.26795i − 0.581748i
\(83\) − 2.19615i − 0.241059i −0.992710 0.120530i \(-0.961541\pi\)
0.992710 0.120530i \(-0.0384592\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0.143594 0.0154841
\(87\) 0 0
\(88\) − 14.5359i − 1.54953i
\(89\) −5.19615 −0.550791 −0.275396 0.961331i \(-0.588809\pi\)
−0.275396 + 0.961331i \(0.588809\pi\)
\(90\) 0 0
\(91\) −6.92820 −0.726273
\(92\) 5.07180i 0.528771i
\(93\) 0 0
\(94\) 6.39230 0.659316
\(95\) 0 0
\(96\) 0 0
\(97\) − 9.66025i − 0.980850i −0.871483 0.490425i \(-0.836842\pi\)
0.871483 0.490425i \(-0.163158\pi\)
\(98\) − 11.2679i − 1.13823i
\(99\) 0 0
\(100\) 0 0
\(101\) −2.66025 −0.264705 −0.132353 0.991203i \(-0.542253\pi\)
−0.132353 + 0.991203i \(0.542253\pi\)
\(102\) 0 0
\(103\) 0.535898i 0.0528036i 0.999651 + 0.0264018i \(0.00840494\pi\)
−0.999651 + 0.0264018i \(0.991595\pi\)
\(104\) 3.71281 0.364071
\(105\) 0 0
\(106\) 4.92820 0.478669
\(107\) − 8.53590i − 0.825196i −0.910913 0.412598i \(-0.864621\pi\)
0.910913 0.412598i \(-0.135379\pi\)
\(108\) 0 0
\(109\) 6.07180 0.581573 0.290786 0.956788i \(-0.406083\pi\)
0.290786 + 0.956788i \(0.406083\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) − 5.07180i − 0.479240i
\(113\) 19.1244i 1.79907i 0.436851 + 0.899534i \(0.356094\pi\)
−0.436851 + 0.899534i \(0.643906\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −4.67949 −0.434480
\(117\) 0 0
\(118\) 6.05256i 0.557183i
\(119\) −12.9282 −1.18513
\(120\) 0 0
\(121\) 21.8564 1.98695
\(122\) 2.92820i 0.265107i
\(123\) 0 0
\(124\) −4.39230 −0.394441
\(125\) 0 0
\(126\) 0 0
\(127\) − 14.5885i − 1.29452i −0.762271 0.647258i \(-0.775916\pi\)
0.762271 0.647258i \(-0.224084\pi\)
\(128\) 10.1436i 0.896575i
\(129\) 0 0
\(130\) 0 0
\(131\) 15.5885 1.36197 0.680985 0.732297i \(-0.261552\pi\)
0.680985 + 0.732297i \(0.261552\pi\)
\(132\) 0 0
\(133\) 21.1244i 1.83171i
\(134\) −2.53590 −0.219068
\(135\) 0 0
\(136\) 6.92820 0.594089
\(137\) 2.53590i 0.216656i 0.994115 + 0.108328i \(0.0345497\pi\)
−0.994115 + 0.108328i \(0.965450\pi\)
\(138\) 0 0
\(139\) −0.607695 −0.0515440 −0.0257720 0.999668i \(-0.508204\pi\)
−0.0257720 + 0.999668i \(0.508204\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) − 2.73205i − 0.229269i
\(143\) 8.39230i 0.701800i
\(144\) 0 0
\(145\) 0 0
\(146\) −5.60770 −0.464096
\(147\) 0 0
\(148\) − 4.00000i − 0.328798i
\(149\) 8.00000 0.655386 0.327693 0.944784i \(-0.393729\pi\)
0.327693 + 0.944784i \(0.393729\pi\)
\(150\) 0 0
\(151\) 5.39230 0.438820 0.219410 0.975633i \(-0.429587\pi\)
0.219410 + 0.975633i \(0.429587\pi\)
\(152\) − 11.3205i − 0.918214i
\(153\) 0 0
\(154\) −19.8564 −1.60007
\(155\) 0 0
\(156\) 0 0
\(157\) − 19.1244i − 1.52629i −0.646227 0.763145i \(-0.723654\pi\)
0.646227 0.763145i \(-0.276346\pi\)
\(158\) − 11.3205i − 0.900611i
\(159\) 0 0
\(160\) 0 0
\(161\) 16.3923 1.29189
\(162\) 0 0
\(163\) − 12.7321i − 0.997251i −0.866817 0.498626i \(-0.833838\pi\)
0.866817 0.498626i \(-0.166162\pi\)
\(164\) −10.5359 −0.822715
\(165\) 0 0
\(166\) 1.60770 0.124781
\(167\) − 17.6603i − 1.36659i −0.730142 0.683296i \(-0.760546\pi\)
0.730142 0.683296i \(-0.239454\pi\)
\(168\) 0 0
\(169\) 10.8564 0.835108
\(170\) 0 0
\(171\) 0 0
\(172\) − 0.287187i − 0.0218978i
\(173\) − 8.53590i − 0.648972i −0.945890 0.324486i \(-0.894809\pi\)
0.945890 0.324486i \(-0.105191\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −6.14359 −0.463091
\(177\) 0 0
\(178\) − 3.80385i − 0.285110i
\(179\) −8.12436 −0.607243 −0.303621 0.952793i \(-0.598196\pi\)
−0.303621 + 0.952793i \(0.598196\pi\)
\(180\) 0 0
\(181\) 26.4641 1.96706 0.983531 0.180742i \(-0.0578498\pi\)
0.983531 + 0.180742i \(0.0578498\pi\)
\(182\) − 5.07180i − 0.375947i
\(183\) 0 0
\(184\) −8.78461 −0.647610
\(185\) 0 0
\(186\) 0 0
\(187\) 15.6603i 1.14519i
\(188\) − 12.7846i − 0.932413i
\(189\) 0 0
\(190\) 0 0
\(191\) −8.12436 −0.587858 −0.293929 0.955827i \(-0.594963\pi\)
−0.293929 + 0.955827i \(0.594963\pi\)
\(192\) 0 0
\(193\) 5.26795i 0.379195i 0.981862 + 0.189598i \(0.0607184\pi\)
−0.981862 + 0.189598i \(0.939282\pi\)
\(194\) 7.07180 0.507725
\(195\) 0 0
\(196\) −22.5359 −1.60971
\(197\) − 13.8564i − 0.987228i −0.869681 0.493614i \(-0.835676\pi\)
0.869681 0.493614i \(-0.164324\pi\)
\(198\) 0 0
\(199\) 2.00000 0.141776 0.0708881 0.997484i \(-0.477417\pi\)
0.0708881 + 0.997484i \(0.477417\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) − 1.94744i − 0.137021i
\(203\) 15.1244i 1.06152i
\(204\) 0 0
\(205\) 0 0
\(206\) −0.392305 −0.0273332
\(207\) 0 0
\(208\) − 1.56922i − 0.108806i
\(209\) 25.5885 1.76999
\(210\) 0 0
\(211\) −8.85641 −0.609700 −0.304850 0.952400i \(-0.598606\pi\)
−0.304850 + 0.952400i \(0.598606\pi\)
\(212\) − 9.85641i − 0.676941i
\(213\) 0 0
\(214\) 6.24871 0.427153
\(215\) 0 0
\(216\) 0 0
\(217\) 14.1962i 0.963698i
\(218\) 4.44486i 0.301044i
\(219\) 0 0
\(220\) 0 0
\(221\) −4.00000 −0.269069
\(222\) 0 0
\(223\) − 16.7846i − 1.12398i −0.827144 0.561990i \(-0.810036\pi\)
0.827144 0.561990i \(-0.189964\pi\)
\(224\) 27.7128 1.85164
\(225\) 0 0
\(226\) −14.0000 −0.931266
\(227\) − 18.0526i − 1.19819i −0.800678 0.599095i \(-0.795527\pi\)
0.800678 0.599095i \(-0.204473\pi\)
\(228\) 0 0
\(229\) 12.0000 0.792982 0.396491 0.918039i \(-0.370228\pi\)
0.396491 + 0.918039i \(0.370228\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) − 8.10512i − 0.532127i
\(233\) 28.0526i 1.83778i 0.394509 + 0.918892i \(0.370915\pi\)
−0.394509 + 0.918892i \(0.629085\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 12.1051 0.787976
\(237\) 0 0
\(238\) − 9.46410i − 0.613467i
\(239\) 0.535898 0.0346644 0.0173322 0.999850i \(-0.494483\pi\)
0.0173322 + 0.999850i \(0.494483\pi\)
\(240\) 0 0
\(241\) −16.3205 −1.05130 −0.525648 0.850702i \(-0.676177\pi\)
−0.525648 + 0.850702i \(0.676177\pi\)
\(242\) 16.0000i 1.02852i
\(243\) 0 0
\(244\) 5.85641 0.374918
\(245\) 0 0
\(246\) 0 0
\(247\) 6.53590i 0.415869i
\(248\) − 7.60770i − 0.483089i
\(249\) 0 0
\(250\) 0 0
\(251\) −10.3923 −0.655956 −0.327978 0.944685i \(-0.606367\pi\)
−0.327978 + 0.944685i \(0.606367\pi\)
\(252\) 0 0
\(253\) − 19.8564i − 1.24836i
\(254\) 10.6795 0.670091
\(255\) 0 0
\(256\) −11.7128 −0.732051
\(257\) 10.3923i 0.648254i 0.946014 + 0.324127i \(0.105071\pi\)
−0.946014 + 0.324127i \(0.894929\pi\)
\(258\) 0 0
\(259\) −12.9282 −0.803319
\(260\) 0 0
\(261\) 0 0
\(262\) 11.4115i 0.705007i
\(263\) − 21.3205i − 1.31468i −0.753595 0.657339i \(-0.771682\pi\)
0.753595 0.657339i \(-0.228318\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −15.4641 −0.948165
\(267\) 0 0
\(268\) 5.07180i 0.309809i
\(269\) −10.6603 −0.649967 −0.324984 0.945720i \(-0.605359\pi\)
−0.324984 + 0.945720i \(0.605359\pi\)
\(270\) 0 0
\(271\) −2.92820 −0.177876 −0.0889378 0.996037i \(-0.528347\pi\)
−0.0889378 + 0.996037i \(0.528347\pi\)
\(272\) − 2.92820i − 0.177548i
\(273\) 0 0
\(274\) −1.85641 −0.112150
\(275\) 0 0
\(276\) 0 0
\(277\) 3.80385i 0.228551i 0.993449 + 0.114276i \(0.0364547\pi\)
−0.993449 + 0.114276i \(0.963545\pi\)
\(278\) − 0.444864i − 0.0266812i
\(279\) 0 0
\(280\) 0 0
\(281\) 15.4641 0.922511 0.461255 0.887267i \(-0.347399\pi\)
0.461255 + 0.887267i \(0.347399\pi\)
\(282\) 0 0
\(283\) 29.3205i 1.74292i 0.490464 + 0.871462i \(0.336827\pi\)
−0.490464 + 0.871462i \(0.663173\pi\)
\(284\) −5.46410 −0.324235
\(285\) 0 0
\(286\) −6.14359 −0.363278
\(287\) 34.0526i 2.01006i
\(288\) 0 0
\(289\) 9.53590 0.560935
\(290\) 0 0
\(291\) 0 0
\(292\) 11.2154i 0.656331i
\(293\) − 28.7321i − 1.67854i −0.543712 0.839272i \(-0.682981\pi\)
0.543712 0.839272i \(-0.317019\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 6.92820 0.402694
\(297\) 0 0
\(298\) 5.85641i 0.339253i
\(299\) 5.07180 0.293310
\(300\) 0 0
\(301\) −0.928203 −0.0535007
\(302\) 3.94744i 0.227150i
\(303\) 0 0
\(304\) −4.78461 −0.274416
\(305\) 0 0
\(306\) 0 0
\(307\) 14.0526i 0.802022i 0.916073 + 0.401011i \(0.131341\pi\)
−0.916073 + 0.401011i \(0.868659\pi\)
\(308\) 39.7128i 2.26285i
\(309\) 0 0
\(310\) 0 0
\(311\) 19.7321 1.11890 0.559451 0.828863i \(-0.311012\pi\)
0.559451 + 0.828863i \(0.311012\pi\)
\(312\) 0 0
\(313\) 9.07180i 0.512768i 0.966575 + 0.256384i \(0.0825312\pi\)
−0.966575 + 0.256384i \(0.917469\pi\)
\(314\) 14.0000 0.790066
\(315\) 0 0
\(316\) −22.6410 −1.27366
\(317\) 6.19615i 0.348011i 0.984745 + 0.174005i \(0.0556710\pi\)
−0.984745 + 0.174005i \(0.944329\pi\)
\(318\) 0 0
\(319\) 18.3205 1.02575
\(320\) 0 0
\(321\) 0 0
\(322\) 12.0000i 0.668734i
\(323\) 12.1962i 0.678612i
\(324\) 0 0
\(325\) 0 0
\(326\) 9.32051 0.516215
\(327\) 0 0
\(328\) − 18.2487i − 1.00762i
\(329\) −41.3205 −2.27807
\(330\) 0 0
\(331\) −0.464102 −0.0255093 −0.0127547 0.999919i \(-0.504060\pi\)
−0.0127547 + 0.999919i \(0.504060\pi\)
\(332\) − 3.21539i − 0.176467i
\(333\) 0 0
\(334\) 12.9282 0.707400
\(335\) 0 0
\(336\) 0 0
\(337\) 27.3205i 1.48824i 0.668044 + 0.744121i \(0.267132\pi\)
−0.668044 + 0.744121i \(0.732868\pi\)
\(338\) 7.94744i 0.432284i
\(339\) 0 0
\(340\) 0 0
\(341\) 17.1962 0.931224
\(342\) 0 0
\(343\) 39.7128i 2.14429i
\(344\) 0.497423 0.0268192
\(345\) 0 0
\(346\) 6.24871 0.335933
\(347\) − 28.5885i − 1.53471i −0.641223 0.767354i \(-0.721573\pi\)
0.641223 0.767354i \(-0.278427\pi\)
\(348\) 0 0
\(349\) 18.8564 1.00936 0.504680 0.863306i \(-0.331610\pi\)
0.504680 + 0.863306i \(0.331610\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) − 33.5692i − 1.78925i
\(353\) − 25.5167i − 1.35811i −0.734085 0.679057i \(-0.762389\pi\)
0.734085 0.679057i \(-0.237611\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −7.60770 −0.403207
\(357\) 0 0
\(358\) − 5.94744i − 0.314332i
\(359\) 6.12436 0.323231 0.161616 0.986854i \(-0.448330\pi\)
0.161616 + 0.986854i \(0.448330\pi\)
\(360\) 0 0
\(361\) 0.928203 0.0488528
\(362\) 19.3731i 1.01823i
\(363\) 0 0
\(364\) −10.1436 −0.531669
\(365\) 0 0
\(366\) 0 0
\(367\) 31.1769i 1.62742i 0.581270 + 0.813711i \(0.302556\pi\)
−0.581270 + 0.813711i \(0.697444\pi\)
\(368\) 3.71281i 0.193544i
\(369\) 0 0
\(370\) 0 0
\(371\) −31.8564 −1.65390
\(372\) 0 0
\(373\) − 20.0526i − 1.03828i −0.854689 0.519141i \(-0.826252\pi\)
0.854689 0.519141i \(-0.173748\pi\)
\(374\) −11.4641 −0.592795
\(375\) 0 0
\(376\) 22.1436 1.14197
\(377\) 4.67949i 0.241006i
\(378\) 0 0
\(379\) −2.39230 −0.122884 −0.0614422 0.998111i \(-0.519570\pi\)
−0.0614422 + 0.998111i \(0.519570\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) − 5.94744i − 0.304298i
\(383\) − 2.53590i − 0.129578i −0.997899 0.0647892i \(-0.979363\pi\)
0.997899 0.0647892i \(-0.0206375\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −3.85641 −0.196286
\(387\) 0 0
\(388\) − 14.1436i − 0.718032i
\(389\) −27.4641 −1.39249 −0.696243 0.717807i \(-0.745146\pi\)
−0.696243 + 0.717807i \(0.745146\pi\)
\(390\) 0 0
\(391\) 9.46410 0.478620
\(392\) − 39.0333i − 1.97148i
\(393\) 0 0
\(394\) 10.1436 0.511027
\(395\) 0 0
\(396\) 0 0
\(397\) 14.3923i 0.722329i 0.932502 + 0.361165i \(0.117621\pi\)
−0.932502 + 0.361165i \(0.882379\pi\)
\(398\) 1.46410i 0.0733888i
\(399\) 0 0
\(400\) 0 0
\(401\) 24.9282 1.24486 0.622428 0.782677i \(-0.286147\pi\)
0.622428 + 0.782677i \(0.286147\pi\)
\(402\) 0 0
\(403\) 4.39230i 0.218796i
\(404\) −3.89488 −0.193778
\(405\) 0 0
\(406\) −11.0718 −0.549484
\(407\) 15.6603i 0.776250i
\(408\) 0 0
\(409\) 17.8564 0.882942 0.441471 0.897275i \(-0.354457\pi\)
0.441471 + 0.897275i \(0.354457\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0.784610i 0.0386549i
\(413\) − 39.1244i − 1.92518i
\(414\) 0 0
\(415\) 0 0
\(416\) 8.57437 0.420393
\(417\) 0 0
\(418\) 18.7321i 0.916215i
\(419\) −20.3923 −0.996229 −0.498115 0.867111i \(-0.665974\pi\)
−0.498115 + 0.867111i \(0.665974\pi\)
\(420\) 0 0
\(421\) 33.7846 1.64656 0.823281 0.567635i \(-0.192141\pi\)
0.823281 + 0.567635i \(0.192141\pi\)
\(422\) − 6.48334i − 0.315604i
\(423\) 0 0
\(424\) 17.0718 0.829080
\(425\) 0 0
\(426\) 0 0
\(427\) − 18.9282i − 0.916000i
\(428\) − 12.4974i − 0.604086i
\(429\) 0 0
\(430\) 0 0
\(431\) −21.3397 −1.02790 −0.513950 0.857820i \(-0.671818\pi\)
−0.513950 + 0.857820i \(0.671818\pi\)
\(432\) 0 0
\(433\) 35.4641i 1.70430i 0.523301 + 0.852148i \(0.324700\pi\)
−0.523301 + 0.852148i \(0.675300\pi\)
\(434\) −10.3923 −0.498847
\(435\) 0 0
\(436\) 8.88973 0.425741
\(437\) − 15.4641i − 0.739748i
\(438\) 0 0
\(439\) −5.39230 −0.257361 −0.128680 0.991686i \(-0.541074\pi\)
−0.128680 + 0.991686i \(0.541074\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) − 2.92820i − 0.139280i
\(443\) 0.339746i 0.0161418i 0.999967 + 0.00807091i \(0.00256908\pi\)
−0.999967 + 0.00807091i \(0.997431\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 12.2872 0.581815
\(447\) 0 0
\(448\) 10.1436i 0.479240i
\(449\) −8.12436 −0.383412 −0.191706 0.981452i \(-0.561402\pi\)
−0.191706 + 0.981452i \(0.561402\pi\)
\(450\) 0 0
\(451\) 41.2487 1.94233
\(452\) 28.0000i 1.31701i
\(453\) 0 0
\(454\) 13.2154 0.620229
\(455\) 0 0
\(456\) 0 0
\(457\) 2.73205i 0.127800i 0.997956 + 0.0639000i \(0.0203539\pi\)
−0.997956 + 0.0639000i \(0.979646\pi\)
\(458\) 8.78461i 0.410478i
\(459\) 0 0
\(460\) 0 0
\(461\) −37.0526 −1.72571 −0.862855 0.505452i \(-0.831326\pi\)
−0.862855 + 0.505452i \(0.831326\pi\)
\(462\) 0 0
\(463\) 10.3923i 0.482971i 0.970404 + 0.241486i \(0.0776347\pi\)
−0.970404 + 0.241486i \(0.922365\pi\)
\(464\) −3.42563 −0.159031
\(465\) 0 0
\(466\) −20.5359 −0.951307
\(467\) 37.3731i 1.72942i 0.502272 + 0.864710i \(0.332498\pi\)
−0.502272 + 0.864710i \(0.667502\pi\)
\(468\) 0 0
\(469\) 16.3923 0.756926
\(470\) 0 0
\(471\) 0 0
\(472\) 20.9667i 0.965070i
\(473\) 1.12436i 0.0516979i
\(474\) 0 0
\(475\) 0 0
\(476\) −18.9282 −0.867573
\(477\) 0 0
\(478\) 0.392305i 0.0179436i
\(479\) 11.8756 0.542612 0.271306 0.962493i \(-0.412544\pi\)
0.271306 + 0.962493i \(0.412544\pi\)
\(480\) 0 0
\(481\) −4.00000 −0.182384
\(482\) − 11.9474i − 0.544191i
\(483\) 0 0
\(484\) 32.0000 1.45455
\(485\) 0 0
\(486\) 0 0
\(487\) 24.3923i 1.10532i 0.833407 + 0.552660i \(0.186387\pi\)
−0.833407 + 0.552660i \(0.813613\pi\)
\(488\) 10.1436i 0.459179i
\(489\) 0 0
\(490\) 0 0
\(491\) −13.8756 −0.626199 −0.313100 0.949720i \(-0.601367\pi\)
−0.313100 + 0.949720i \(0.601367\pi\)
\(492\) 0 0
\(493\) 8.73205i 0.393272i
\(494\) −4.78461 −0.215270
\(495\) 0 0
\(496\) −3.21539 −0.144375
\(497\) 17.6603i 0.792171i
\(498\) 0 0
\(499\) −24.3205 −1.08874 −0.544368 0.838847i \(-0.683230\pi\)
−0.544368 + 0.838847i \(0.683230\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 7.60770i − 0.339548i
\(503\) 7.32051i 0.326405i 0.986593 + 0.163203i \(0.0521824\pi\)
−0.986593 + 0.163203i \(0.947818\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 14.5359 0.646200
\(507\) 0 0
\(508\) − 21.3590i − 0.947652i
\(509\) 6.78461 0.300723 0.150361 0.988631i \(-0.451956\pi\)
0.150361 + 0.988631i \(0.451956\pi\)
\(510\) 0 0
\(511\) 36.2487 1.60355
\(512\) 11.7128i 0.517638i
\(513\) 0 0
\(514\) −7.60770 −0.335561
\(515\) 0 0
\(516\) 0 0
\(517\) 50.0526i 2.20131i
\(518\) − 9.46410i − 0.415829i
\(519\) 0 0
\(520\) 0 0
\(521\) 19.4641 0.852738 0.426369 0.904549i \(-0.359793\pi\)
0.426369 + 0.904549i \(0.359793\pi\)
\(522\) 0 0
\(523\) − 22.2487i − 0.972868i −0.873717 0.486434i \(-0.838297\pi\)
0.873717 0.486434i \(-0.161703\pi\)
\(524\) 22.8231 0.997031
\(525\) 0 0
\(526\) 15.6077 0.680528
\(527\) 8.19615i 0.357030i
\(528\) 0 0
\(529\) 11.0000 0.478261
\(530\) 0 0
\(531\) 0 0
\(532\) 30.9282i 1.34091i
\(533\) 10.5359i 0.456360i
\(534\) 0 0
\(535\) 0 0
\(536\) −8.78461 −0.379437
\(537\) 0 0
\(538\) − 7.80385i − 0.336448i
\(539\) 88.2295 3.80031
\(540\) 0 0
\(541\) −24.4641 −1.05179 −0.525897 0.850548i \(-0.676270\pi\)
−0.525897 + 0.850548i \(0.676270\pi\)
\(542\) − 2.14359i − 0.0920752i
\(543\) 0 0
\(544\) 16.0000 0.685994
\(545\) 0 0
\(546\) 0 0
\(547\) − 33.8564i − 1.44760i −0.690012 0.723798i \(-0.742395\pi\)
0.690012 0.723798i \(-0.257605\pi\)
\(548\) 3.71281i 0.158604i
\(549\) 0 0
\(550\) 0 0
\(551\) 14.2679 0.607835
\(552\) 0 0
\(553\) 73.1769i 3.11180i
\(554\) −2.78461 −0.118307
\(555\) 0 0
\(556\) −0.889727 −0.0377328
\(557\) − 2.53590i − 0.107449i −0.998556 0.0537247i \(-0.982891\pi\)
0.998556 0.0537247i \(-0.0171094\pi\)
\(558\) 0 0
\(559\) −0.287187 −0.0121467
\(560\) 0 0
\(561\) 0 0
\(562\) 11.3205i 0.477527i
\(563\) 16.7321i 0.705172i 0.935779 + 0.352586i \(0.114698\pi\)
−0.935779 + 0.352586i \(0.885302\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −21.4641 −0.902203
\(567\) 0 0
\(568\) − 9.46410i − 0.397105i
\(569\) 32.9090 1.37962 0.689808 0.723993i \(-0.257695\pi\)
0.689808 + 0.723993i \(0.257695\pi\)
\(570\) 0 0
\(571\) −39.7846 −1.66493 −0.832467 0.554075i \(-0.813072\pi\)
−0.832467 + 0.554075i \(0.813072\pi\)
\(572\) 12.2872i 0.513753i
\(573\) 0 0
\(574\) −24.9282 −1.04048
\(575\) 0 0
\(576\) 0 0
\(577\) 15.2679i 0.635613i 0.948156 + 0.317807i \(0.102946\pi\)
−0.948156 + 0.317807i \(0.897054\pi\)
\(578\) 6.98076i 0.290361i
\(579\) 0 0
\(580\) 0 0
\(581\) −10.3923 −0.431145
\(582\) 0 0
\(583\) 38.5885i 1.59817i
\(584\) −19.4256 −0.803838
\(585\) 0 0
\(586\) 21.0333 0.868878
\(587\) − 11.6603i − 0.481270i −0.970616 0.240635i \(-0.922644\pi\)
0.970616 0.240635i \(-0.0773557\pi\)
\(588\) 0 0
\(589\) 13.3923 0.551820
\(590\) 0 0
\(591\) 0 0
\(592\) − 2.92820i − 0.120348i
\(593\) 0.143594i 0.00589668i 0.999996 + 0.00294834i \(0.000938487\pi\)
−0.999996 + 0.00294834i \(0.999062\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 11.7128 0.479776
\(597\) 0 0
\(598\) 3.71281i 0.151828i
\(599\) −27.1962 −1.11120 −0.555602 0.831448i \(-0.687512\pi\)
−0.555602 + 0.831448i \(0.687512\pi\)
\(600\) 0 0
\(601\) −31.2487 −1.27466 −0.637331 0.770590i \(-0.719961\pi\)
−0.637331 + 0.770590i \(0.719961\pi\)
\(602\) − 0.679492i − 0.0276940i
\(603\) 0 0
\(604\) 7.89488 0.321238
\(605\) 0 0
\(606\) 0 0
\(607\) − 16.1962i − 0.657382i −0.944438 0.328691i \(-0.893393\pi\)
0.944438 0.328691i \(-0.106607\pi\)
\(608\) − 26.1436i − 1.06026i
\(609\) 0 0
\(610\) 0 0
\(611\) −12.7846 −0.517210
\(612\) 0 0
\(613\) − 1.46410i − 0.0591345i −0.999563 0.0295673i \(-0.990587\pi\)
0.999563 0.0295673i \(-0.00941292\pi\)
\(614\) −10.2872 −0.415157
\(615\) 0 0
\(616\) −68.7846 −2.77141
\(617\) − 6.92820i − 0.278919i −0.990228 0.139459i \(-0.955464\pi\)
0.990228 0.139459i \(-0.0445365\pi\)
\(618\) 0 0
\(619\) 11.8564 0.476549 0.238275 0.971198i \(-0.423418\pi\)
0.238275 + 0.971198i \(0.423418\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 14.4449i 0.579186i
\(623\) 24.5885i 0.985116i
\(624\) 0 0
\(625\) 0 0
\(626\) −6.64102 −0.265428
\(627\) 0 0
\(628\) − 28.0000i − 1.11732i
\(629\) −7.46410 −0.297613
\(630\) 0 0
\(631\) −32.7128 −1.30228 −0.651138 0.758959i \(-0.725708\pi\)
−0.651138 + 0.758959i \(0.725708\pi\)
\(632\) − 39.2154i − 1.55990i
\(633\) 0 0
\(634\) −4.53590 −0.180144
\(635\) 0 0
\(636\) 0 0
\(637\) 22.5359i 0.892905i
\(638\) 13.4115i 0.530968i
\(639\) 0 0
\(640\) 0 0
\(641\) 9.33975 0.368898 0.184449 0.982842i \(-0.440950\pi\)
0.184449 + 0.982842i \(0.440950\pi\)
\(642\) 0 0
\(643\) − 41.6603i − 1.64292i −0.570266 0.821460i \(-0.693160\pi\)
0.570266 0.821460i \(-0.306840\pi\)
\(644\) 24.0000 0.945732
\(645\) 0 0
\(646\) −8.92820 −0.351275
\(647\) 11.4641i 0.450700i 0.974278 + 0.225350i \(0.0723526\pi\)
−0.974278 + 0.225350i \(0.927647\pi\)
\(648\) 0 0
\(649\) −47.3923 −1.86031
\(650\) 0 0
\(651\) 0 0
\(652\) − 18.6410i − 0.730039i
\(653\) 17.4641i 0.683423i 0.939805 + 0.341712i \(0.111007\pi\)
−0.939805 + 0.341712i \(0.888993\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −7.71281 −0.301135
\(657\) 0 0
\(658\) − 30.2487i − 1.17922i
\(659\) 1.46410 0.0570333 0.0285167 0.999593i \(-0.490922\pi\)
0.0285167 + 0.999593i \(0.490922\pi\)
\(660\) 0 0
\(661\) 18.3205 0.712585 0.356293 0.934374i \(-0.384041\pi\)
0.356293 + 0.934374i \(0.384041\pi\)
\(662\) − 0.339746i − 0.0132046i
\(663\) 0 0
\(664\) 5.56922 0.216128
\(665\) 0 0
\(666\) 0 0
\(667\) − 11.0718i − 0.428702i
\(668\) − 25.8564i − 1.00041i
\(669\) 0 0
\(670\) 0 0
\(671\) −22.9282 −0.885133
\(672\) 0 0
\(673\) 10.3923i 0.400594i 0.979735 + 0.200297i \(0.0641907\pi\)
−0.979735 + 0.200297i \(0.935809\pi\)
\(674\) −20.0000 −0.770371
\(675\) 0 0
\(676\) 15.8949 0.611342
\(677\) − 18.0000i − 0.691796i −0.938272 0.345898i \(-0.887574\pi\)
0.938272 0.345898i \(-0.112426\pi\)
\(678\) 0 0
\(679\) −45.7128 −1.75430
\(680\) 0 0
\(681\) 0 0
\(682\) 12.5885i 0.482037i
\(683\) 19.6077i 0.750268i 0.926971 + 0.375134i \(0.122403\pi\)
−0.926971 + 0.375134i \(0.877597\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −29.0718 −1.10997
\(687\) 0 0
\(688\) − 0.210236i − 0.00801515i
\(689\) −9.85641 −0.375499
\(690\) 0 0
\(691\) −17.7128 −0.673827 −0.336914 0.941536i \(-0.609383\pi\)
−0.336914 + 0.941536i \(0.609383\pi\)
\(692\) − 12.4974i − 0.475081i
\(693\) 0 0
\(694\) 20.9282 0.794424
\(695\) 0 0
\(696\) 0 0
\(697\) 19.6603i 0.744685i
\(698\) 13.8038i 0.522483i
\(699\) 0 0
\(700\) 0 0
\(701\) −20.8038 −0.785750 −0.392875 0.919592i \(-0.628520\pi\)
−0.392875 + 0.919592i \(0.628520\pi\)
\(702\) 0 0
\(703\) 12.1962i 0.459987i
\(704\) 12.2872 0.463091
\(705\) 0 0
\(706\) 18.6795 0.703012
\(707\) 12.5885i 0.473438i
\(708\) 0 0
\(709\) 22.5359 0.846353 0.423177 0.906047i \(-0.360915\pi\)
0.423177 + 0.906047i \(0.360915\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) − 13.1769i − 0.493826i
\(713\) − 10.3923i − 0.389195i
\(714\) 0 0
\(715\) 0 0
\(716\) −11.8949 −0.444533
\(717\) 0 0
\(718\) 4.48334i 0.167317i
\(719\) −8.41154 −0.313698 −0.156849 0.987623i \(-0.550134\pi\)
−0.156849 + 0.987623i \(0.550134\pi\)
\(720\) 0 0
\(721\) 2.53590 0.0944418
\(722\) 0.679492i 0.0252881i
\(723\) 0 0
\(724\) 38.7461 1.43999
\(725\) 0 0
\(726\) 0 0
\(727\) − 8.39230i − 0.311253i −0.987816 0.155627i \(-0.950260\pi\)
0.987816 0.155627i \(-0.0497397\pi\)
\(728\) − 17.5692i − 0.651159i
\(729\) 0 0
\(730\) 0 0
\(731\) −0.535898 −0.0198209
\(732\) 0 0
\(733\) − 34.7846i − 1.28480i −0.766370 0.642399i \(-0.777939\pi\)
0.766370 0.642399i \(-0.222061\pi\)
\(734\) −22.8231 −0.842415
\(735\) 0 0
\(736\) −20.2872 −0.747796
\(737\) − 19.8564i − 0.731420i
\(738\) 0 0
\(739\) −22.4641 −0.826355 −0.413178 0.910650i \(-0.635581\pi\)
−0.413178 + 0.910650i \(0.635581\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) − 23.3205i − 0.856123i
\(743\) − 19.9090i − 0.730389i −0.930931 0.365195i \(-0.881002\pi\)
0.930931 0.365195i \(-0.118998\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 14.6795 0.537454
\(747\) 0 0
\(748\) 22.9282i 0.838338i
\(749\) −40.3923 −1.47590
\(750\) 0 0
\(751\) 48.7846 1.78018 0.890088 0.455789i \(-0.150643\pi\)
0.890088 + 0.455789i \(0.150643\pi\)
\(752\) − 9.35898i − 0.341287i
\(753\) 0 0
\(754\) −3.42563 −0.124754
\(755\) 0 0
\(756\) 0 0
\(757\) − 9.17691i − 0.333541i −0.985996 0.166770i \(-0.946666\pi\)
0.985996 0.166770i \(-0.0533338\pi\)
\(758\) − 1.75129i − 0.0636097i
\(759\) 0 0
\(760\) 0 0
\(761\) 23.4449 0.849876 0.424938 0.905223i \(-0.360296\pi\)
0.424938 + 0.905223i \(0.360296\pi\)
\(762\) 0 0
\(763\) − 28.7321i − 1.04017i
\(764\) −11.8949 −0.430342
\(765\) 0 0
\(766\) 1.85641 0.0670747
\(767\) − 12.1051i − 0.437090i
\(768\) 0 0
\(769\) 9.53590 0.343873 0.171937 0.985108i \(-0.444998\pi\)
0.171937 + 0.985108i \(0.444998\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 7.71281i 0.277590i
\(773\) − 1.51666i − 0.0545505i −0.999628 0.0272752i \(-0.991317\pi\)
0.999628 0.0272752i \(-0.00868306\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 24.4974 0.879406
\(777\) 0 0
\(778\) − 20.1051i − 0.720803i
\(779\) 32.1244 1.15097
\(780\) 0 0
\(781\) 21.3923 0.765477
\(782\) 6.92820i 0.247752i
\(783\) 0 0
\(784\) −16.4974 −0.589194
\(785\) 0 0
\(786\) 0 0
\(787\) − 48.0526i − 1.71289i −0.516239 0.856444i \(-0.672668\pi\)
0.516239 0.856444i \(-0.327332\pi\)
\(788\) − 20.2872i − 0.722701i
\(789\) 0 0
\(790\) 0 0
\(791\) 90.4974 3.21772
\(792\) 0 0
\(793\) − 5.85641i − 0.207967i
\(794\) −10.5359 −0.373905
\(795\) 0 0
\(796\) 2.92820 0.103787
\(797\) − 15.6077i − 0.552853i −0.961035 0.276426i \(-0.910850\pi\)
0.961035 0.276426i \(-0.0891502\pi\)
\(798\) 0 0
\(799\) −23.8564 −0.843979
\(800\) 0 0
\(801\) 0 0
\(802\) 18.2487i 0.644384i
\(803\) − 43.9090i − 1.54951i
\(804\) 0 0
\(805\) 0 0
\(806\) −3.21539 −0.113257
\(807\) 0 0
\(808\) − 6.74613i − 0.237328i
\(809\) −45.4449 −1.59776 −0.798878 0.601493i \(-0.794573\pi\)
−0.798878 + 0.601493i \(0.794573\pi\)
\(810\) 0 0
\(811\) −18.4641 −0.648362 −0.324181 0.945995i \(-0.605089\pi\)
−0.324181 + 0.945995i \(0.605089\pi\)
\(812\) 22.1436i 0.777088i
\(813\) 0 0
\(814\) −11.4641 −0.401817
\(815\) 0 0
\(816\) 0 0
\(817\) 0.875644i 0.0306349i
\(818\) 13.0718i 0.457045i
\(819\) 0 0
\(820\) 0 0
\(821\) −37.7321 −1.31686 −0.658429 0.752643i \(-0.728779\pi\)
−0.658429 + 0.752643i \(0.728779\pi\)
\(822\) 0 0
\(823\) − 3.85641i − 0.134426i −0.997739 0.0672129i \(-0.978589\pi\)
0.997739 0.0672129i \(-0.0214107\pi\)
\(824\) −1.35898 −0.0473424
\(825\) 0 0
\(826\) 28.6410 0.996548
\(827\) − 11.6077i − 0.403639i −0.979423 0.201820i \(-0.935315\pi\)
0.979423 0.201820i \(-0.0646855\pi\)
\(828\) 0 0
\(829\) 23.7846 0.826074 0.413037 0.910714i \(-0.364468\pi\)
0.413037 + 0.910714i \(0.364468\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 3.13844i 0.108806i
\(833\) 42.0526i 1.45703i
\(834\) 0 0
\(835\) 0 0
\(836\) 37.4641 1.29572
\(837\) 0 0
\(838\) − 14.9282i − 0.515686i
\(839\) −33.1962 −1.14606 −0.573029 0.819535i \(-0.694232\pi\)
−0.573029 + 0.819535i \(0.694232\pi\)
\(840\) 0 0
\(841\) −18.7846 −0.647745
\(842\) 24.7321i 0.852323i
\(843\) 0 0
\(844\) −12.9667 −0.446331
\(845\) 0 0
\(846\) 0 0
\(847\) − 103.426i − 3.55375i
\(848\) − 7.21539i − 0.247778i
\(849\) 0 0
\(850\) 0 0
\(851\) 9.46410 0.324425
\(852\) 0 0
\(853\) − 10.4833i − 0.358943i −0.983763 0.179471i \(-0.942561\pi\)
0.983763 0.179471i \(-0.0574387\pi\)
\(854\) 13.8564 0.474156
\(855\) 0 0
\(856\) 21.6462 0.739851
\(857\) 44.4974i 1.52000i 0.649921 + 0.760002i \(0.274802\pi\)
−0.649921 + 0.760002i \(0.725198\pi\)
\(858\) 0 0
\(859\) 11.0000 0.375315 0.187658 0.982235i \(-0.439910\pi\)
0.187658 + 0.982235i \(0.439910\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) − 15.6218i − 0.532080i
\(863\) 23.1244i 0.787162i 0.919290 + 0.393581i \(0.128764\pi\)
−0.919290 + 0.393581i \(0.871236\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −25.9615 −0.882209
\(867\) 0 0
\(868\) 20.7846i 0.705476i
\(869\) 88.6410 3.00694
\(870\) 0 0
\(871\) 5.07180 0.171851
\(872\) 15.3975i 0.521424i
\(873\) 0 0
\(874\) 11.3205 0.382922
\(875\) 0 0
\(876\) 0 0
\(877\) 33.4641i 1.13000i 0.825090 + 0.565001i \(0.191124\pi\)
−0.825090 + 0.565001i \(0.808876\pi\)
\(878\) − 3.94744i − 0.133220i
\(879\) 0 0
\(880\) 0 0
\(881\) −47.0526 −1.58524 −0.792620 0.609715i \(-0.791284\pi\)
−0.792620 + 0.609715i \(0.791284\pi\)
\(882\) 0 0
\(883\) 30.1962i 1.01618i 0.861304 + 0.508091i \(0.169649\pi\)
−0.861304 + 0.508091i \(0.830351\pi\)
\(884\) −5.85641 −0.196972
\(885\) 0 0
\(886\) −0.248711 −0.00835562
\(887\) 33.8038i 1.13502i 0.823366 + 0.567511i \(0.192094\pi\)
−0.823366 + 0.567511i \(0.807906\pi\)
\(888\) 0 0
\(889\) −69.0333 −2.31530
\(890\) 0 0
\(891\) 0 0
\(892\) − 24.5744i − 0.822811i
\(893\) 38.9808i 1.30444i
\(894\) 0 0
\(895\) 0 0
\(896\) 48.0000 1.60357
\(897\) 0 0
\(898\) − 5.94744i − 0.198469i
\(899\) 9.58846 0.319793
\(900\) 0 0
\(901\) −18.3923 −0.612737
\(902\) 30.1962i 1.00542i
\(903\) 0 0
\(904\) −48.4974 −1.61300
\(905\) 0 0
\(906\) 0 0
\(907\) − 30.0000i − 0.996134i −0.867139 0.498067i \(-0.834043\pi\)
0.867139 0.498067i \(-0.165957\pi\)
\(908\) − 26.4308i − 0.877136i
\(909\) 0 0
\(910\) 0 0
\(911\) 23.5885 0.781520 0.390760 0.920493i \(-0.372212\pi\)
0.390760 + 0.920493i \(0.372212\pi\)
\(912\) 0 0
\(913\) 12.5885i 0.416617i
\(914\) −2.00000 −0.0661541
\(915\) 0 0
\(916\) 17.5692 0.580503
\(917\) − 73.7654i − 2.43595i
\(918\) 0 0
\(919\) 56.9615 1.87899 0.939494 0.342566i \(-0.111296\pi\)
0.939494 + 0.342566i \(0.111296\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) − 27.1244i − 0.893293i
\(923\) 5.46410i 0.179853i
\(924\) 0 0
\(925\) 0 0
\(926\) −7.60770 −0.250004
\(927\) 0 0
\(928\) − 18.7180i − 0.614447i
\(929\) 44.3731 1.45583 0.727917 0.685666i \(-0.240489\pi\)
0.727917 + 0.685666i \(0.240489\pi\)
\(930\) 0 0
\(931\) 68.7128 2.25197
\(932\) 41.0718i 1.34535i
\(933\) 0 0
\(934\) −27.3590 −0.895213
\(935\) 0 0
\(936\) 0 0
\(937\) − 4.14359i − 0.135365i −0.997707 0.0676827i \(-0.978439\pi\)
0.997707 0.0676827i \(-0.0215605\pi\)
\(938\) 12.0000i 0.391814i
\(939\) 0 0
\(940\) 0 0
\(941\) 35.1769 1.14673 0.573367 0.819298i \(-0.305637\pi\)
0.573367 + 0.819298i \(0.305637\pi\)
\(942\) 0 0
\(943\) − 24.9282i − 0.811774i
\(944\) 8.86156 0.288419
\(945\) 0 0
\(946\) −0.823085 −0.0267608
\(947\) − 57.7128i − 1.87541i −0.347427 0.937707i \(-0.612944\pi\)
0.347427 0.937707i \(-0.387056\pi\)
\(948\) 0 0
\(949\) 11.2154 0.364067
\(950\) 0 0
\(951\) 0 0
\(952\) − 32.7846i − 1.06256i
\(953\) − 36.3923i − 1.17886i −0.807819 0.589431i \(-0.799352\pi\)
0.807819 0.589431i \(-0.200648\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0.784610 0.0253761
\(957\) 0 0
\(958\) 8.69358i 0.280877i
\(959\) 12.0000 0.387500
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) − 2.92820i − 0.0944091i
\(963\) 0 0
\(964\) −23.8949 −0.769602
\(965\) 0 0
\(966\) 0 0
\(967\) − 25.8038i − 0.829796i −0.909868 0.414898i \(-0.863817\pi\)
0.909868 0.414898i \(-0.136183\pi\)
\(968\) 55.4256i 1.78145i
\(969\) 0 0
\(970\) 0 0
\(971\) 17.4449 0.559832 0.279916 0.960024i \(-0.409693\pi\)
0.279916 + 0.960024i \(0.409693\pi\)
\(972\) 0 0
\(973\) 2.87564i 0.0921889i
\(974\) −17.8564 −0.572156
\(975\) 0 0
\(976\) 4.28719 0.137230
\(977\) 5.46410i 0.174812i 0.996173 + 0.0874060i \(0.0278578\pi\)
−0.996173 + 0.0874060i \(0.972142\pi\)
\(978\) 0 0
\(979\) 29.7846 0.951920
\(980\) 0 0
\(981\) 0 0
\(982\) − 10.1577i − 0.324144i
\(983\) − 48.5885i − 1.54973i −0.632126 0.774866i \(-0.717818\pi\)
0.632126 0.774866i \(-0.282182\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −6.39230 −0.203572
\(987\) 0 0
\(988\) 9.56922i 0.304437i
\(989\) 0.679492 0.0216066
\(990\) 0 0
\(991\) 30.8564 0.980186 0.490093 0.871670i \(-0.336963\pi\)
0.490093 + 0.871670i \(0.336963\pi\)
\(992\) − 17.5692i − 0.557823i
\(993\) 0 0
\(994\) −12.9282 −0.410058
\(995\) 0 0
\(996\) 0 0
\(997\) 25.5167i 0.808121i 0.914732 + 0.404060i \(0.132401\pi\)
−0.914732 + 0.404060i \(0.867599\pi\)
\(998\) − 17.8038i − 0.563571i
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2025.2.b.g.649.3 4
3.2 odd 2 2025.2.b.h.649.2 4
5.2 odd 4 2025.2.a.m.1.1 2
5.3 odd 4 405.2.a.g.1.2 2
5.4 even 2 inner 2025.2.b.g.649.2 4
15.2 even 4 2025.2.a.g.1.2 2
15.8 even 4 405.2.a.h.1.1 yes 2
15.14 odd 2 2025.2.b.h.649.3 4
20.3 even 4 6480.2.a.br.1.2 2
45.13 odd 12 405.2.e.l.136.1 4
45.23 even 12 405.2.e.i.136.2 4
45.38 even 12 405.2.e.i.271.2 4
45.43 odd 12 405.2.e.l.271.1 4
60.23 odd 4 6480.2.a.bi.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
405.2.a.g.1.2 2 5.3 odd 4
405.2.a.h.1.1 yes 2 15.8 even 4
405.2.e.i.136.2 4 45.23 even 12
405.2.e.i.271.2 4 45.38 even 12
405.2.e.l.136.1 4 45.13 odd 12
405.2.e.l.271.1 4 45.43 odd 12
2025.2.a.g.1.2 2 15.2 even 4
2025.2.a.m.1.1 2 5.2 odd 4
2025.2.b.g.649.2 4 5.4 even 2 inner
2025.2.b.g.649.3 4 1.1 even 1 trivial
2025.2.b.h.649.2 4 3.2 odd 2
2025.2.b.h.649.3 4 15.14 odd 2
6480.2.a.bi.1.2 2 60.23 odd 4
6480.2.a.br.1.2 2 20.3 even 4